Groups with free regular length functions in $\mathbb{Z}^{n}$

Groups with free regular length functions in Zn

This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on Zn-trees give one a powerful tool to study groups. All finitely generated groups acting freely on R-trees also act freely on some Zn-trees, but the latter ones form a much larger class. The natural ...

Comparing Length Functions on Free Groups

The aim of this note is to study a pair of length functions on a free group, associated to a point in the unprojectivized Outer space. We discuss various characteristics which compare them and describe how these may be unified by a “Manhattan curve” analogous to those used to study pairs of hyperbolic surfaces. Our approach involves the thermodynamic formalism used in the ergodic theory of hype...

Regular sets and counting in free groups

Regular symmetric groups of boolean functions

is called a Boolean function. By Aut(f) we denote the set of all symmetries of f , i.e., these permutation σ ∈ Sn for which f(xσ(1), . . . , xσ(n)) = f(x1, . . . , xn). We show the solution of a problem posed by A. Kisielewicz ([1]). We show that, with the exception of four known groups of small order, every regular permutation group is isomorphic with Aut(f) for some Boolean function f . We pr...

On Torsion-free Groups with Finite Regular File Bases

The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free group G has a finite system of generators a1, . . . , an such that every element of G has a unique presentation in the form a k1 1 · · · a kn n where ki ∈ Z, is it true that G is virtually polycyclic? The answer is “not always.” A counterexample is constructed in this paper as a group presented ...